🪡K-Theory Unit 5 – The Chern Character and Chern Classes

Key Concepts and Definitions

• Chern classes are characteristic classes associated with complex vector bundles, measuring their non-triviality and providing invariants
• The Chern character is a ring homomorphism from the K-theory of a topological space $X$ to the cohomology ring of $X$ with rational coefficients
  • Connects K-theory and cohomology, allowing for the study of vector bundles using cohomological tools
• K-theory is a generalized cohomology theory that studies vector bundles over topological spaces
  • Provides a framework for understanding the topological properties of vector bundles
• Characteristic classes are cohomology classes associated with vector bundles, capturing their global topological properties
• The total Chern class $c(E)$ of a complex vector bundle $E$ is the formal sum of all Chern classes, $c(E) = 1 + c_1(E) + c_2(E) + \cdots$
• The splitting principle states that any complex vector bundle can be pulled back to a sum of line bundles, simplifying computations
• The Chern character satisfies the Whitney sum formula, $\operatorname{ch}(E \oplus F) = \operatorname{ch}(E) + \operatorname{ch}(F)$, for vector bundles $E$ and $F$

Historical Context and Development

• Chern classes were introduced by Shiing-Shen Chern in the 1940s as a generalization of the Euler class for complex vector bundles
  • Chern's work built upon the earlier contributions of Élie Cartan and Hermann Weyl in differential geometry and topology
• The development of K-theory by Alexander Grothendieck in the 1950s provided a new framework for studying vector bundles
  • Grothendieck's work revolutionized algebraic geometry and had far-reaching consequences in topology
• The Chern character was introduced by Chern in the 1950s as a means of connecting K-theory and cohomology
  • This connection allowed for the application of powerful cohomological tools to the study of vector bundles
• The Atiyah-Singer index theorem, proved by Michael Atiyah and Isadore Singer in the 1960s, highlighted the importance of the Chern character in relating topological and analytical invariants
• The development of equivariant K-theory and the equivariant Chern character in the 1960s and 1970s expanded the scope of these tools to spaces with group actions
• The work of Daniel Quillen in the 1970s on higher algebraic K-theory further generalized the notion of K-theory and the Chern character
• Recent developments, such as the introduction of twisted K-theory and the twisted Chern character, continue to expand the reach and applicability of these concepts

Chern Classes: Foundations

• Chern classes are cohomology classes associated with complex vector bundles, capturing their topological properties
• The $i$-th Chern class $c_i(E)$ of a complex vector bundle $E$ is an element of the $2i$-th cohomology group of the base space
  • $c_i(E) \in H^{2i}(X; \mathbb{Z})$, where $X$ is the base space of the vector bundle
• Chern classes are natural under pullbacks, meaning that if $f: Y \to X$ is a continuous map and $E$ is a vector bundle over $X$, then $c_i(f^*E) = f^*c_i(E)$
• The first Chern class $c_1(E)$ classifies complex line bundles up to isomorphism
  • Two complex line bundles are isomorphic if and only if their first Chern classes are equal
• The top Chern class $c_n(E)$ of a rank $n$ vector bundle $E$ is equal to the Euler class of the underlying real vector bundle
• Chern classes satisfy the Whitney sum formula, $c(E \oplus F) = c(E) \smile c(F)$, where $\smile$ denotes the cup product in cohomology
• The splitting principle allows for the computation of Chern classes by reducing a vector bundle to a sum of line bundles
  • For a rank $n$ vector bundle $E$, the splitting principle yields $c(E) = (1 + x_1) \cdots (1 + x_n)$, where $x_i$ are the Chern roots

The Chern Character: Introduction

• The Chern character is a ring homomorphism from the K-theory of a topological space $X$ to the cohomology ring of $X$ with rational coefficients
  • $\operatorname{ch}: K(X) \to H^*(X; \mathbb{Q})$, where $K(X)$ is the K-theory ring of $X$ and $H^*(X; \mathbb{Q})$ is the cohomology ring with rational coefficients
• For a complex vector bundle $E$, the Chern character is defined as $\operatorname{ch}(E) = \operatorname{tr}(\exp(iF/2\pi))$, where $F$ is the curvature of a connection on $E$
  • This definition is independent of the choice of connection and yields a well-defined cohomology class
• The Chern character satisfies the Whitney sum formula, $\operatorname{ch}(E \oplus F) = \operatorname{ch}(E) + \operatorname{ch}(F)$, for vector bundles $E$ and $F$
• The Chern character is multiplicative under tensor products, $\operatorname{ch}(E \otimes F) = \operatorname{ch}(E) \smile \operatorname{ch}(F)$, where $\smile$ denotes the cup product
• The Chern character can be expressed in terms of Chern classes using the formal identity $\operatorname{ch}(E) = \operatorname{rank}(E) + c_1(E) + \frac{1}{2}(c_1(E)^2 - 2c_2(E)) + \cdots$
• The Chern character is a rational isomorphism between K-theory and cohomology, allowing for the study of vector bundles using cohomological tools
  • This isomorphism is crucial in the proof of the Atiyah-Singer index theorem

Connections to Vector Bundles

• Chern classes and the Chern character are intrinsically linked to the study of complex vector bundles
• The Chern classes of a vector bundle provide a complete set of invariants, determining the bundle up to stable equivalence
  • Two vector bundles are stably equivalent if they become isomorphic after adding trivial bundles
• The Chern character of a vector bundle encodes its Chern classes in a compact form, facilitating computations and connections to cohomology
• The splitting principle allows for the computation of Chern classes and the Chern character by reducing a vector bundle to a sum of line bundles
  • This reduction simplifies calculations and provides a geometric interpretation of these invariants
• The Chern character establishes a connection between K-theory and cohomology, enabling the use of cohomological tools in the study of vector bundles
  • This connection is particularly useful in the context of the Atiyah-Singer index theorem and its applications
• The study of characteristic classes, including Chern classes, provides insight into the global topological properties of vector bundles
  • These properties are essential in understanding the geometry and topology of the base space
• The Chern character and Chern classes play a crucial role in the classification of vector bundles, particularly in the context of algebraic geometry and complex manifolds

Computational Techniques

• The splitting principle is a powerful tool for computing Chern classes and the Chern character
  • It allows for the reduction of a vector bundle to a sum of line bundles, simplifying calculations
• For a rank $n$ vector bundle $E$, the splitting principle yields $c(E) = (1 + x_1) \cdots (1 + x_n)$, where $x_i$ are the Chern roots
  • The Chern character can then be computed as $\operatorname{ch}(E) = e^{x_1} + \cdots + e^{x_n}$
• The Whitney sum formula, $c(E \oplus F) = c(E) \smile c(F)$ and $\operatorname{ch}(E \oplus F) = \operatorname{ch}(E) + \operatorname{ch}(F)$, simplifies computations for direct sums of vector bundles
• The multiplicative property of the Chern character under tensor products, $\operatorname{ch}(E \otimes F) = \operatorname{ch}(E) \smile \operatorname{ch}(F)$, allows for the computation of the Chern character of tensor products
• The Grothendieck-Riemann-Roch theorem relates the Chern character of a pushforward bundle to the Chern character of the original bundle and the Todd class of the tangent bundle
  • This theorem is a powerful tool for computing Chern characters in algebraic geometry and complex manifolds
• The Atiyah-Singer index theorem expresses the index of an elliptic operator in terms of the Chern character of the symbol bundle and the Todd class of the tangent bundle
  • This theorem provides a means of computing indices using topological data
• In practice, the computation of Chern classes and the Chern character often involves the use of algebraic methods, such as the Chern-Weil theory and the Grothendieck-Riemann-Roch theorem

Applications in Topology and Geometry

• Chern classes and the Chern character have numerous applications in topology and geometry, providing invariants and tools for studying vector bundles and their base spaces
• In algebraic geometry, Chern classes are used to define the Chow ring of a smooth projective variety
  • The Chow ring is a fundamental invariant that captures the algebraic cycles on the variety
• The Chern character is a key ingredient in the Hirzebruch-Riemann-Roch theorem, which relates the Euler characteristic of a coherent sheaf to its Chern character
  • This theorem is a powerful tool for computing invariants of algebraic varieties
• In complex geometry, Chern classes are used to define the Chern numbers of a complex manifold
  • Chern numbers are important invariants that capture the global properties of the manifold
• The Atiyah-Singer index theorem, which relies on the Chern character, has applications in the study of elliptic operators and their indices
  • This theorem connects analysis, topology, and geometry, providing a means of computing indices using topological data
• The Chern character plays a crucial role in the Grothendieck-Riemann-Roch theorem, which relates the Chern character of a pushforward bundle to the Chern character of the original bundle and the Todd class of the tangent bundle
  • This theorem has applications in the study of algebraic cycles and the intersection theory of algebraic varieties
• In topology, Chern classes and the Chern character are used to study the K-theory of topological spaces and its relation to cohomology
  • This connection provides a powerful framework for understanding the topological properties of vector bundles and their base spaces

Advanced Topics and Current Research

• Equivariant K-theory and the equivariant Chern character extend the notions of K-theory and the Chern character to spaces with group actions
  • These equivariant versions provide tools for studying vector bundles and their invariants in the presence of symmetries
• Higher algebraic K-theory, developed by Daniel Quillen, generalizes the notion of K-theory to higher categories
  • The higher Chern character connects higher algebraic K-theory to cyclic homology, providing a means of studying more sophisticated invariants
• Twisted K-theory and the twisted Chern character incorporate the notion of twisting by a cohomology class or a gerbe
  • These twisted versions have applications in string theory and the study of D-branes
• The Chern character and its generalizations play a crucial role in the study of index theory and the geometry of elliptic operators
  • Current research in this area focuses on extending the Atiyah-Singer index theorem to more general settings and exploring its connections to physics
• The Chern character has been generalized to the setting of noncommutative geometry, where it plays a key role in the study of noncommutative spaces and their K-theory
  • Noncommutative geometry has applications in quantum field theory and the geometry of quantum groups
• In algebraic geometry, the study of Chern classes and the Chern character has been extended to the setting of derived categories and derived algebraic geometry
  • These generalizations provide new tools for understanding the geometry of algebraic varieties and their invariants
• Current research in the area of Chern classes and the Chern character focuses on their applications in various fields, such as string theory, mirror symmetry, and the geometry of moduli spaces